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Chapter 8, Solution 6. (a)
Let i = the inductor current. For t < 0, u(t) = 0 so that i(0) = 0 and v(0) = 0. For t > 0, u(t) = 1. Since, v(0+) = v(0-) = 0, and i(0+) = i(0-) = 0. v R (0+) = Ri(0+) = 0 V Also, since v(0+) = v R (0+) + v L (0+) = 0 = 0 + v L (0+) or v L (0+) = 0 V. (1)
(b)
Since i(0+) = 0, But,
i C (0+) = V S /R S
i C = Cdv/dt which leads to dv(0+)/dt = V S /(CR S ) (2)
From (1), (3)
dv(0+)/dt = dv R (0+)/dt + dv L (0+)/dt v R = iR or dv R /dt = Rdi/dt
(4) But,
v L = Ldi/dt, v L (0+) = 0 = Ldi(0+)/dt and di(0+)/dt = 0
(5)
From (4) and (5),
dv R (0+)/dt = 0 V/s
From (2) and (3),
dv L (0+)/dt = dv(0+)/dt = V s /(CR s )
(c) As t approaches infinity, the capacitor acts like an open circuit, while the inductor acts like a short circuit. v R () = [R/(R + R s )]V s v L () = 0 V
